Introduction:
In the last few years there have been significant advances
in the characterization of the mechanical properties of carbon nanotubes,
silicon and silicon oxide surfaces and cantilevers [1-4]. Properties such as resonance frequency and
bending modulus have been extensively researched for both cantilevers and
carbon nanotubes [1, 5-8].
A great deal of work has also been done on the mechanical properties of
functionalized cantilevers, generally covered on one face with a metallic layer
of variable thickness and composition, polymer coatings, or monolayers of
self-assembled or chemically bonded molecules of various types [8-16]. Functionalized cantilevers exhibit
deflection due to differences in surface stress between functionalized and
non-functionalized opposite faces. The
surface stress on the functionalized face is a function of the environment, and
the changes induced by the environment are generally reversible. Since it is possible to control the
environment, it is also possible to generate controlled deflection of the
cantilever. This presents significant opportunities to utilize functionalized
cantilevers as reliable and reversible actuators in nanomechanical devices such
as valves, pumps, switches, etc.
In this paper we present the design of a fluid control
valve that utilizes a silicon cantilever, functionalized with a covalently
bonded monolayer of acrylic acid, as the actuator that opens and closes flow
through a fluid conduit, a single-wall carbon nanotube (SWCNT). The on/off position of the valve is
controlled by pH changes in the surrounding environment. Changes in pH affect the charge of the
organic acid groups bonded to the surface of the cantilever. The electrostatic energy of these acid
groups on the functionalized surface of the cantilever causes a compressive
stress that deflects it to the closed position [10]. The device assembly and valve components are
feasible with today's laboratory synthesis capabilities (SWCNT synthesis
methods, silicon etching techniques and covalent monolayer assembly).
Classical engineering design approximations can be utilized
to lower the computational costs of current molecular modeling methodologies.
As devices become larger, their design becomes computationally more expensive
and in some cases impractical (an N2 problem). In the present case, for example, the system
has in excess of 75,000 atoms and is over 30 nm long. Since the performance of the system depends
on the electrostatic interaction of all the functional molecules on the surface
of the device, it is necessary to include in the calculations all the
electrostatic interactions of all charged particles in order to have an
accurate model. Efficient electrostatic
lattice sum methods, such as Ewald and Particle-Mesh Ewald, cannot be employed
without introducing artifacts due to imposition of periodic boundary conditions
(the device under consideration is not a periodic system), thus we are left
with a direct electrostatic sum in real space.
In most standard molecular simulation packages this would require
including all non-bonded interactions within a radius of 35 nm. Since the
number of non-bonded interactions scales as N2, this makes the
calculations unnecessarily lengthy assuming that the computing system has
enough memory to store such a large energy expression. Furthermore, the use of cutoffs or splines
in the calculation of the electrostatic energy can underestimate the correct
values by factors of one order of magnitude for the monolayer dimensions
considered here.
A second computational issue is the statistical nature of
the evaluation of the convergence criteria used in a molecular mechanics
simulation. Average forces or strain
energies don't necessarily represent the equilibrium state of the system, local
or global. Due to the large number of
atoms, large residual forces and stresses may be present in a small section of
the system, while the average force and strain are quite small. A criteria based on average forces/strains
may leave the device in a non-equilibrium state.
These difficulties will be overcome with more powerful
computer systems and more flexible molecular simulations software packages, but
this requires time, human and financial resources, and breakthroughs in
computer systems. A more practical
approach is to perform a semi-continuum characterization of the system, whereby
each component of a particular device is individually characterized using
molecular simulations prior to a classical analysis of the entire device. This characterization includes the
determination of the classical engineering parameters needed for the continuum
analysis, such as natural vibrational frequencies, elasticity moduli, points of
mechanical failure, etc., at the length scale under consideration. Once these parameters are available, the
continuum analysis of the assembled device becomes a simple exercise. The latter approach is presented in this
paper.
It must also be noted that in order to have a complete ab
initio description of the device presented here, there are several fundamental
issues that still need to be answered.
Two of them are the molecular transport phenomena through SWCNT's and
the solvation properties of acid monolayers on deflected cantilevers. Although no attempt is made to resolve these
issues in this paper, the engineering method presented here remains a useful
design tool for the future inclusion of these effects. This is because it is
generally possible to find smooth-varying mathematical approximations of those
effects, over finite length and time scales, which can then be incorporated
into the engineering continuum models.
Atomistic simulations provide the range over which these smooth
approximations are valid as well as identify the regions where transitions take
place. A prime example of a complex
system which can be classically approximated with segment-wise smoothly varying
functions is the bending of a SWCNT, which exhibits buckling phenomena (see figure
9).
Figure
9: 17,17 carbon nanotube at a curvature of
0.0031/Ang (beyond the point of buckling).